====================================================================================================
GRAPH K_2   n=2 d=1 diam=1 bipartite=True
 lambda_2=-1.0 (mult m2=1)  lambda_n=-1.0   tau(0)=0.5   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.0 at p=11/40 ; min tau/boundB = 1.0000500050005 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 0.999900009999 at p=9999/10000 ; max/tau(0) = 1.9998 ; max/d = 0.99990001
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=1
 T2 numeric tau(1-10^-k)-d: k=1:-0.090909 k=2:-0.009901 k=3:-0.000999 k=4:-9.999e-5 k=6:-1.0e-6 k=8:-1.0e-8 k=10:-1.0e-10 k=12:-1.0e-12 k=16:-1.0e-16 k=20:-1.0e-20 k=24:-1.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = 1.0e-24
 T4: m2=1  n*f0(e)^2=1.0  Gamma(f0)=2.0  c0=(m2-1)/2+(n/4)Gamma=1.0   s*=max over V=1.0  (s*-c0=-7.7788e-62)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.0025125628 | -3.1504e-59 | -4.66726e-61 | tau_abs(0)=inf, tau_abs(p)=100.0
     3 : 0.00025012506 | -2.33752e-58 | 7.7321e-59 | tau_abs(0)=inf, tau_abs(p)=1000.0
     4 : 2.500125e-5 | -8.56054e-58 | 6.99622e-58 | tau_abs(0)=inf, tau_abs(p)=10000.0
     5 : 2.5000125e-6 | -3.19711e-56 | 1.47014e-56 | tau_abs(0)=inf, tau_abs(p)=100000.0
     6 : 2.5000013e-7 | -1.87547e-55 | -3.19713e-56 | tau_abs(0)=inf, tau_abs(p)=1000000.0
     8 : 2.5e-9 | -5.30832e-53 | 9.14698e-54 | tau_abs(0)=inf, tau_abs(p)=1.0e+8
    10 : 2.5e-11 | -2.08659e-52 | -2.08659e-52 | tau_abs(0)=inf, tau_abs(p)=1.0e+10
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.0s)
====================================================================================================
GRAPH K_3   n=3 d=2 diam=1 bipartite=False
 lambda_2=-0.5 (mult m2=2)  lambda_n=-0.5   tau(0)=0.666666666667   tau_abs(0)=2.0
 T3 on 45 values of p: min tau/boundA = 1.0 at p=39/40 ; min tau/boundB = 1.500100010001 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 1.99980001 at p=9999/10000 ; max/tau(0) = 2.9997 ; max/d = 0.9999
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=1
 T2 numeric tau(1-10^-k)-d: k=1:-0.19048 k=2:-0.0199 k=3:-0.001999 k=4:-0.00019999 k=6:-2.0e-6 k=8:-2.0e-8 k=10:-2.0e-10 k=12:-2.0e-12 k=16:-2.0e-16 k=20:-2.0e-20 k=24:-2.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = -3.1115e-61
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=2.0  c0=(m2-1)/2+(n/4)Gamma=2.0   s*=max over V=2.0  (s*-c0=0.0)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.0089186176 | -0.019802 | -0.019802 | 0.0
     3 : 0.00088918528 | -0.001998 | -0.001998 | -1.2446031e-60
     4 : 8.8891852e-5 | -0.00019998 | -0.00019998 | 0.0
     5 : 8.8889185e-6 | -1.99998e-5 | -1.99998e-5 | 6.2230153e-61
     6 : 8.8888919e-7 | -2.0e-6 | -2.0e-6 | 0.0
     8 : 8.8888889e-9 | -2.0e-8 | -2.0e-8 | 3.1115076e-61
    10 : 8.8888889e-11 | -2.0e-10 | -2.0e-10 | 3.1115076e-61
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [0, 0, 0, 0, 0, 0, 0]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.0s)
====================================================================================================
GRAPH K_4   n=4 d=3 diam=1 bipartite=False
 lambda_2=-0.333333333333 (mult m2=3)  lambda_n=-0.333333333333   tau(0)=0.75   tau_abs(0)=1.5
 T3 on 45 values of p: min tau/boundA = 1.0 at p=999/1000 ; min tau/boundB = 2.0001500150015 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 2.99970001 at p=9999/10000 ; max/tau(0) = 3.9996 ; max/d = 0.9999
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=1
 T2 numeric tau(1-10^-k)-d: k=1:-0.29032 k=2:-0.0299 k=3:-0.002999 k=4:-0.00029999 k=6:-3.0e-6 k=8:-3.0e-8 k=10:-3.0e-10 k=12:-3.0e-12 k=16:-3.0e-16 k=20:-3.0e-20 k=24:-3.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = -3.0e-24
 T4: m2=3  n*f0(e)^2=3.0  Gamma(f0)=2.0  c0=(m2-1)/2+(n/4)Gamma=3.0   s*=max over V=3.0  (s*-c0=3.1115e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.016917293 | -0.0588235 | -0.0588235 | -1.5557538e-61
     3 : 0.001687922 | -0.00598802 | -0.00598802 | 3.1115076e-61
     4 : 0.00016875422 | -0.00059988 | -0.00059988 | 3.1115076e-61
     5 : 1.6875042e-5 | -5.99988e-5 | -5.99988e-5 | -1.5557538e-61
     6 : 1.6875004e-6 | -5.99999e-6 | -5.99999e-6 | 3.1115076e-61
     8 : 1.6875e-8 | -6.0e-8 | -6.0e-8 | 4.6672615e-61
    10 : 1.6875e-10 | -6.0e-10 | -6.0e-10 | -1.5557538e-61
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [0, 0, 0, 0, 0, 0, 0]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.0s)
====================================================================================================
GRAPH K_5   n=5 d=4 diam=1 bipartite=False
 lambda_2=-0.25 (mult m2=4)  lambda_n=-0.25   tau(0)=0.8   tau_abs(0)=1.33333333333
 T3 on 45 values of p: min tau/boundA = 1.0 at p=999/1000 ; min tau/boundB = 2.500200020002 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 3.99960001 at p=9999/10000 ; max/tau(0) = 4.9995 ; max/d = 0.9999
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=1
 T2 numeric tau(1-10^-k)-d: k=1:-0.39024 k=2:-0.0399 k=3:-0.003999 k=4:-0.00039999 k=6:-4.0e-6 k=8:-4.0e-8 k=10:-4.0e-10 k=12:-4.0e-12 k=16:-4.0e-16 k=20:-4.0e-20 k=24:-4.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = -4.0e-24
 T4: m2=4  n*f0(e)^2=4.0  Gamma(f0)=2.0  c0=(m2-1)/2+(n/4)Gamma=4.0   s*=max over V=4.0  (s*-c0=-9.3345e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.025651303 | -0.116505 | -0.116505 | 0.0
     3 : 0.0025605121 | -0.0119641 | -0.0119641 | 0.0
     4 : 0.00025600512 | -0.00119964 | -0.00119964 | 1.5557538e-61
     5 : 2.5600051e-5 | -0.000119996 | -0.000119996 | 0.0
     6 : 2.5600005e-6 | -1.2e-5 | -1.2e-5 | 3.1115076e-61
     8 : 2.56e-8 | -1.2e-7 | -1.2e-7 | 0.0
    10 : 2.56e-10 | -1.2e-9 | -1.2e-9 | 3.1115076e-61
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [0, 0, 0, 0, 0, 0, 0]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.1s)
====================================================================================================
GRAPH K_6   n=6 d=5 diam=1 bipartite=False
 lambda_2=-0.2 (mult m2=5)  lambda_n=-0.2   tau(0)=0.833333333333   tau_abs(0)=1.25
 T3 on 45 values of p: min tau/boundA = 1.0 at p=37/40 ; min tau/boundB = 3.0002500250025 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 4.99950001 at p=9999/10000 ; max/tau(0) = 5.9994 ; max/d = 0.9999
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=1
 T2 numeric tau(1-10^-k)-d: k=1:-0.4902 k=2:-0.0499 k=3:-0.004999 k=4:-0.00049999 k=6:-5.0e-6 k=8:-5.0e-8 k=10:-5.0e-10 k=12:-5.0e-12 k=16:-5.0e-16 k=20:-5.0e-20 k=24:-5.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = -5.0e-24
 T4: m2=5  n*f0(e)^2=5.0  Gamma(f0)=2.0  c0=(m2-1)/2+(n/4)Gamma=5.0   s*=max over V=5.0  (s*-c0=-1.2446e-60)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.034780189 | -0.192308 | -0.192308 | 3.1115076e-61
     3 : 0.003472801 | -0.0199203 | -0.0199203 | 0.0
     4 : 0.00034722801 | -0.0019992 | -0.0019992 | 1.5557538e-61
     5 : 3.472228e-5 | -0.000199992 | -0.000199992 | 0.0
     6 : 3.4722228e-6 | -1.99999e-5 | -1.99999e-5 | 1.5557538e-61
     8 : 3.4722222e-8 | -2.0e-7 | -2.0e-7 | 3.1115076e-61
    10 : 3.4722222e-10 | -2.0e-9 | -2.0e-9 | 3.1115076e-61
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [0, 0, 0, 0, 0, 0, 0]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.1s)
====================================================================================================
GRAPH K_7   n=7 d=6 diam=1 bipartite=False
 lambda_2=-0.166666666667 (mult m2=6)  lambda_n=-0.166666666667   tau(0)=0.857142857143   tau_abs(0)=1.2
 T3 on 45 values of p: min tau/boundA = 1.0 at p=39/40 ; min tau/boundB = 3.500300030003 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 5.99940001 at p=9999/10000 ; max/tau(0) = 6.9993 ; max/d = 0.9999
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=1
 T2 numeric tau(1-10^-k)-d: k=1:-0.59016 k=2:-0.0599 k=3:-0.005999 k=4:-0.00059999 k=6:-6.0e-6 k=8:-6.0e-8 k=10:-6.0e-10 k=12:-6.0e-12 k=16:-6.0e-16 k=20:-6.0e-20 k=24:-6.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = -6.0e-24
 T4: m2=6  n*f0(e)^2=6.0  Gamma(f0)=2.0  c0=(m2-1)/2+(n/4)Gamma=6.0   s*=max over V=6.0  (s*-c0=0.0)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.044144697 | -0.285714 | -0.285714 | 0.0
     3 : 0.0044087931 | -0.0298507 | -0.0298507 | 0.0
     4 : 0.00044082262 | -0.0029985 | -0.0029985 | 3.1115076e-61
     5 : 4.4081696e-5 | -0.000299985 | -0.000299985 | 1.5557538e-61
     6 : 4.4081639e-6 | -2.99999e-5 | -2.99999e-5 | 3.1115076e-61
     8 : 4.4081633e-8 | -3.0e-7 | -3.0e-7 | 1.5557538e-61
    10 : 4.4081633e-10 | -3.0e-9 | -3.0e-9 | 1.5557538e-61
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [0, 0, 0, 0, 0, 0, 0]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.1s)
====================================================================================================
GRAPH K_8   n=8 d=7 diam=1 bipartite=False
 lambda_2=-0.142857142857 (mult m2=7)  lambda_n=-0.142857142857   tau(0)=0.875   tau_abs(0)=1.16666666667
 T3 on 45 values of p: min tau/boundA = 1.0 at p=19/20 ; min tau/boundB = 4.0003500350035 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 6.99930001 at p=9999/10000 ; max/tau(0) = 7.9992 ; max/d = 0.9999
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=1
 T2 numeric tau(1-10^-k)-d: k=1:-0.69014 k=2:-0.0699 k=3:-0.006999 k=4:-0.00069999 k=6:-7.0e-6 k=8:-7.0e-8 k=10:-7.0e-10 k=12:-7.0e-12 k=16:-7.0e-16 k=20:-7.0e-20 k=24:-7.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = -7.0e-24
 T4: m2=7  n*f0(e)^2=7.0  Gamma(f0)=2.0  c0=(m2-1)/2+(n/4)Gamma=7.0   s*=max over V=7.0  (s*-c0=3.7338e-60)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.053660826 | -0.396226 | -0.396226 | -1.5557538e-61
     3 : 0.005360045 | -0.0417495 | -0.0417495 | 0.0
     4 : 0.0005359442 | -0.00419748 | -0.00419748 | 0.0
     5 : 5.3593817e-5 | -0.000419975 | -0.000419975 | -1.5557538e-61
     6 : 5.3593757e-6 | -4.19997e-5 | -4.19997e-5 | 1.5557538e-61
     8 : 5.359375e-8 | -4.2e-7 | -4.2e-7 | -1.5557538e-61
    10 : 5.359375e-10 | -4.2e-9 | -4.2e-9 | 0.0
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [0, 0, 0, 0, 0, 0, 0]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.1s)
====================================================================================================
GRAPH Z_3   n=3 d=2 diam=1 bipartite=False
 lambda_2=-0.5 (mult m2=2)  lambda_n=-0.5   tau(0)=0.666666666667   tau_abs(0)=2.0
 T3 on 45 values of p: min tau/boundA = 1.0 at p=39/40 ; min tau/boundB = 1.500100010001 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 1.99980001 at p=9999/10000 ; max/tau(0) = 2.9997 ; max/d = 0.9999
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=1
 T2 numeric tau(1-10^-k)-d: k=1:-0.19048 k=2:-0.0199 k=3:-0.001999 k=4:-0.00019999 k=6:-2.0e-6 k=8:-2.0e-8 k=10:-2.0e-10 k=12:-2.0e-12 k=16:-2.0e-16 k=20:-2.0e-20 k=24:-2.0e-24
    |tau-d| ~ eps^1.0 (from k=16,24); tau_abs(1-10^-24)-d = -3.1115e-61
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=2.0  c0=(m2-1)/2+(n/4)Gamma=2.0   s*=max over V=2.0  (s*-c0=0.0)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.0089186176 | -0.019802 | -0.019802 | 0.0
     3 : 0.00088918528 | -0.001998 | -0.001998 | -1.2446031e-60
     4 : 8.8891852e-5 | -0.00019998 | -0.00019998 | 0.0
     5 : 8.8889185e-6 | -1.99998e-5 | -1.99998e-5 | 6.2230153e-61
     6 : 8.8888919e-7 | -2.0e-6 | -2.0e-6 | 0.0
     8 : 8.8888889e-9 | -2.0e-8 | -2.0e-8 | 3.1115076e-61
    10 : 8.8888889e-11 | -2.0e-10 | -2.0e-10 | 3.1115076e-61
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [0, 0, 0, 0, 0, 0, 0]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.0s)
====================================================================================================
GRAPH Z_4   n=4 d=2 diam=2 bipartite=True
 lambda_2=9.07016631075e-62 (mult m2=2)  lambda_n=-1.0   tau(0)=1.0   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.00004999250137 at p=9999/10000 ; min tau/boundB = 1.00010001499875 at p=1/10000 ; violations=0
 shape on grid: increasing ; max tau = 1.99999998 at p=9999/10000 ; max/tau(0) = 2.0 ; max/d = 0.99999999
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=2
 T2 numeric tau(1-10^-k)-d: k=1:-0.017556 k=2:-0.00019707 k=3:-1.997e-6 k=4:-1.9997e-8 k=6:-2.0e-12 k=8:-2.0e-16 k=10:-2.0e-20 k=12:-2.0e-24 k=16:-2.0e-32 k=20:-2.0e-40 k=24:-2.0e-48
    |tau-d| ~ eps^2.0 (from k=16,24); tau_abs(1-10^-24)-d = -2.0e-48
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=1.0  c0=(m2-1)/2+(n/4)Gamma=1.5   s*=max over V=1.5  (s*-c0=3.1115e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.015022913 | -0.0199435 | -0.0246064 | tau_abs(0)=inf, tau_abs(p)=51.239918
     3 : 0.0015002479 | -0.00199949 | -0.00249601 | tau_abs(0)=inf, tau_abs(p)=501.249
     4 : 0.0001500025 | -0.000199995 | -0.00024996 | tau_abs(0)=inf, tau_abs(p)=5001.2499
     5 : 1.5000025e-5 | -1.99999e-5 | -2.49996e-5 | tau_abs(0)=inf, tau_abs(p)=50001.25
     6 : 1.5000002e-6 | -2.0e-6 | -2.5e-6 | tau_abs(0)=inf, tau_abs(p)=500001.25
     8 : 1.5e-8 | -2.0e-8 | -2.5e-8 | tau_abs(0)=inf, tau_abs(p)=50000001.0
    10 : 1.5e-10 | -2.0e-10 | -2.5e-10 | tau_abs(0)=inf, tau_abs(p)=5.0e+9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.0s)
====================================================================================================
GRAPH Z_5   n=5 d=2 diam=2 bipartite=False
 lambda_2=0.309016994375 (mult m2=2)  lambda_n=-0.809016994375   tau(0)=1.4472135955   tau_abs(0)=5.2360679775
 T3 on 45 values of p: min tau/boundA = 1.00709593271939 at p=9999/10000 ; min tau/boundB = 1.29226577760011 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 2.44976870881 at p=29/40 ; max/tau(0) = 1.6927485 ; max/d = 1.2248844
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=3
 T2 numeric tau(1-10^-k)-d: k=1:0.36463 k=2:0.13543 k=3:0.04419 k=4:0.014091 k=6:0.0014137 k=8:0.00014142 k=10:1.4142e-5 k=12:1.4142e-6 k=16:1.4142e-8 k=20:1.4142e-10 k=24:1.4142e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 1.4142e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.705572809  c0=(m2-1)/2+(n/4)Gamma=1.38196601125011   s*=max over V=1.38196601125011  (s*-c0=4.6673e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.028723716 | -0.0372206 | -0.0383265 | -0.18887225
     3 : 0.0028922455 | -0.00379593 | -0.00392429 | -0.019739288
     4 : 0.00028942093 | -0.000380342 | -0.00039337 | -0.0019828713
     5 : 2.8944054e-5 | -3.80417e-5 | -3.93465e-5 | -0.000198377
     6 : 2.894425e-6 | -3.80425e-6 | -3.93474e-6 | -1.9838599e-5
     8 : 2.8944272e-8 | -3.80426e-8 | -3.93475e-8 | -1.9838698e-7
    10 : 2.8944272e-10 | -3.80426e-10 | -3.93475e-10 | -1.9838699e-9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [-1, -1, -1, -1, -1, -1, -1]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.1s)
====================================================================================================
GRAPH Z_6   n=6 d=2 diam=3 bipartite=True
 lambda_2=0.5 (mult m2=2)  lambda_n=-1.0   tau(0)=2.0   tau_abs(0)=inf
 T3 on 45 values of p: min tau/boundA = 1.00714646177652 at p=9999/10000 ; min tau/boundB = 1.09107605197573 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 2.94667549068 at p=19/40 ; max/tau(0) = 1.4733377 ; max/d = 1.4733377
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=4
 T2 numeric tau(1-10^-k)-d: k=1:0.48857 k=2:0.1464 k=3:0.045223 k=4:0.014192 k=6:0.0014147 k=8:0.00014143 k=10:1.4142e-5 k=12:1.4142e-6 k=16:1.4142e-8 k=20:1.4142e-10 k=24:1.4142e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 1.4142e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.5  c0=(m2-1)/2+(n/4)Gamma=1.25   s*=max over V=1.25  (s*-c0=3.1115e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.049062354 | -0.0528096 | -0.0532755 | tau_abs(0)=inf, tau_abs(p)=35.552586
     3 : 0.0049905869 | -0.00545877 | -0.00551947 | tau_abs(0)=inf, tau_abs(p)=335.58863
     4 : 0.00049990584 | -0.000547712 | -0.000553944 | tau_abs(0)=inf, tau_abs(p)=3335.5922
     5 : 4.9999058e-5 | -5.47896e-5 | -5.54144e-5 | tau_abs(0)=inf, tau_abs(p)=33335.593
     6 : 4.9999906e-6 | -5.47915e-6 | -5.54164e-6 | tau_abs(0)=inf, tau_abs(p)=333335.59
     8 : 4.9999999e-8 | -5.47917e-8 | -5.54167e-8 | tau_abs(0)=inf, tau_abs(p)=33333336.0
    10 : 5.0e-10 | -5.47917e-10 | -5.54167e-10 | tau_abs(0)=inf, tau_abs(p)=3.3333333e+9
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [None, None, None, None, None, None, None]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.1s)
====================================================================================================
GRAPH Z_7   n=7 d=2 diam=3 bipartite=False
 lambda_2=0.623489801859 (mult m2=2)  lambda_n=-0.900968867902   tau(0)=2.65597055521   tau_abs(0)=10.097834679
 T3 on 45 values of p: min tau/boundA = 1.01008791397668 at p=9999/10000 ; min tau/boundB = 1.25155484082171 at p=1/10000 ; violations=0
 shape on grid: up-then-down ; max tau = 3.65295591617 at p=7/20 ; max/tau(0) = 1.3753752 ; max/d = 1.826478
 T2 exact limit matrix P_1: lower_triangular=True, absorbing_at_e=True, B1_has_eig_1-1/d=True, row_sums_ok=True, all_eigs_real=True, roots_above_1-1/d_in_blocks_k>=1=0, mult_of_1-1/d=5
 T2 numeric tau(1-10^-k)-d: k=1:0.69774 k=2:0.20729 k=3:0.06399 k=4:0.020075 k=6:0.0020007 k=8:0.00020001 k=10:2.0e-5 k=12:2.0e-6 k=16:2.0e-8 k=20:2.0e-10 k=24:2.0e-12
    |tau-d| ~ eps^0.5 (from k=16,24); tau_abs(1-10^-24)-d = 2.0e-12
 T4: m2=2  n*f0(e)^2=2.0  Gamma(f0)=0.378292198729  c0=(m2-1)/2+(n/4)Gamma=1.16201134777643   s*=max over V=1.16201134777643  (s*-c0=4.6673e-61)
    k : tau(p)-tau(0) | [1/tau(0)-1/tau(p)]/p - s* | [R_0(f0)-R_p(f0)]/p - c0 | tau_abs(p)-tau_abs(0)   (p=10^-k)
     2 : 0.079314126 | -0.0702574 | -0.07028 | -1.2063885
     3 : 0.0081700625 | -0.00737565 | -0.00738279 | -0.13908703
     4 : 0.00081943353 | -0.000741242 | -0.000742027 | -0.014125221
     5 : 8.1967665e-5 | -7.41612e-5 | -7.42404e-5 | -0.0014147249
     6 : 8.1970097e-6 | -7.41649e-6 | -7.42441e-6 | -0.00014149456
     8 : 8.1970365e-8 | -7.41653e-8 | -7.42446e-8 | -1.4149698e-6
    10 : 8.1970367e-10 | -7.41653e-10 | -7.42446e-10 | -1.4149701e-8
    sign pattern of tau_abs(p)-tau_abs(0) for k=2,3,4,5,6,8,10: [-1, -1, -1, -1, -1, -1, -1]
 FLAGS: lemma1a_at_e=True lemma1c_mu(e)>mu(x)=True T3a_holds_on_grid=True T3b_holds_on_grid=True T2_exact=True T2_numeric_limit_d=True T2_abs_numeric_limit_d=True T4_n_f0(e)^2=m2=True T4_c0>0=True T4_sign_tau(p)>tau(0)_p=1e-2..1e-10=True T4_slope_matches_sstar=True T4_Rayleigh_slope_matches_c0=True T4_sstar>=c0=True
 RESULT: ALL OK   (0.2s)

SUMMARY (name | n | d | bip | m2 | tau(0) | shape | max/tau0 | c0 | s* | minA | minB | tau(1-1e-24)-d | exponent | status | abs-sign)
 K_2                                |   2 |  1 | B |  1 | 0.5 | increasing   | 1.9998 | 1.0 | 1.0 | 1.0 | 1.00005 | -1.0e-24 | 1.0 | OK | [None, None, None, None, None, None, None]
 K_3                                |   3 |  2 | - |  2 | 0.66666667 | increasing   | 2.9997 | 2.0 | 2.0 | 1.0 | 1.5001 | -2.0e-24 | 1.0 | OK | [0, 0, 0, 0, 0, 0, 0]
 K_4                                |   4 |  3 | - |  3 | 0.75 | increasing   | 3.9996 | 3.0 | 3.0 | 1.0 | 2.00015 | -3.0e-24 | 1.0 | OK | [0, 0, 0, 0, 0, 0, 0]
 K_5                                |   5 |  4 | - |  4 | 0.8 | increasing   | 4.9995 | 4.0 | 4.0 | 1.0 | 2.5002 | -4.0e-24 | 1.0 | OK | [0, 0, 0, 0, 0, 0, 0]
 K_6                                |   6 |  5 | - |  5 | 0.83333333 | increasing   | 5.9994 | 5.0 | 5.0 | 1.0 | 3.00025 | -5.0e-24 | 1.0 | OK | [0, 0, 0, 0, 0, 0, 0]
 K_7                                |   7 |  6 | - |  6 | 0.85714286 | increasing   | 6.9993 | 6.0 | 6.0 | 1.0 | 3.5003 | -6.0e-24 | 1.0 | OK | [0, 0, 0, 0, 0, 0, 0]
 K_8                                |   8 |  7 | - |  7 | 0.875 | increasing   | 7.9992 | 7.0 | 7.0 | 1.0 | 4.00035 | -7.0e-24 | 1.0 | OK | [0, 0, 0, 0, 0, 0, 0]
 Z_3                                |   3 |  2 | - |  2 | 0.66666667 | increasing   | 2.9997 | 2.0 | 2.0 | 1.0 | 1.5001 | -2.0e-24 | 1.0 | OK | [0, 0, 0, 0, 0, 0, 0]
 Z_4                                |   4 |  2 | B |  2 | 1.0 | increasing   | 2.0 | 1.5 | 1.5 | 1.00005 | 1.0001 | -2.0e-48 | 2.0 | OK | [None, None, None, None, None, None, None]
 Z_5                                |   5 |  2 | - |  2 | 1.4472136 | up-then-down | 1.69275 | 1.381966 | 1.381966 | 1.0070959 | 1.2922658 | 1.41e-12 | 0.5 | OK | [-1, -1, -1, -1, -1, -1, -1]
 Z_6                                |   6 |  2 | B |  2 | 2.0 | up-then-down | 1.47334 | 1.25 | 1.25 | 1.0071465 | 1.0910761 | 1.41e-12 | 0.5 | OK | [None, None, None, None, None, None, None]
 Z_7                                |   7 |  2 | - |  2 | 2.6559706 | up-then-down | 1.37538 | 1.1620113 | 1.1620113 | 1.0100879 | 1.2515548 | 2.0e-12 | 0.5 | OK | [-1, -1, -1, -1, -1, -1, -1]
